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liquid sodium turbulent flow
François Pétrélis, Mickaël Bourgoin, Louis Marie, Javier Burguete, Arnaud Chiffaudel, François Daviaud, Stéphan Fauve, Philippe Odier, Jean-François
Pinton
To cite this version:
François Pétrélis, Mickaël Bourgoin, Louis Marie, Javier Burguete, Arnaud Chiffaudel, et al.. Non- linear magnetic induction by helical motion in a liquid sodium turbulent flow. Physical Review Letters, American Physical Society, 2003, 90 (17), pp.174501. �10.1103/PhysRevLett.90.174501�. �hal- 00492370�
Nonlinear Magnetic Induction by Helical Motion in a Liquid Sodium Turbulent Flow
F. Pe´tre´lis,1M. Bourgoin,2L. Marie´,3J. Burguete,3,* A. Chiffaudel,3F. Daviaud,3S. Fauve,1P. Odier,2and J.-F. Pinton2
1Laboratoire de Physique Statistique de l’Ecole Normale Supe´rieure, CNRS UMR 8550, 24 Rue Lhomond, 75231 Paris Cedex 05, France
2Laboratoire de Physique de l’Ecole Normale Supe´rieure de Lyon, CNRS UMR 5672, 47 alle´e d’Italie, 69364 Lyon Cedex 07, France
3Service de Physique de l’Etat Condense´, Direction des Sciences de la Matie`re, CEA-Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette cedex, France
(Received 7 December 2001; revised manuscript received 14 January 2003; published 2 May 2003) We report an experimental study of the magnetic field BB~ induced by a turbulent swirling flow of liquid sodium submitted to a transverse magnetic fieldBB~0. We show that the induced field can behave nonlinearly as a function of the magnetic Reynolds number,Rm. At lowRm, the induced mean field along the axis of the flow,hBxi, and the one parallel toBB~0,hByi, first behave likeR2m, whereas the third component,hBzi, is linear inRm. The sign ofhBxiis determined by the flow helicity. At higherRm,BB~ strongly depends on the local geometry of the mean flow:hBxi decreases to zero in the core of the swirling flow but remains finite outside. We compare the experimental results with the computed magnetic induction due to the mean flow alone.
DOI: 10.1103/PhysRevLett.90.174501 PACS numbers: 47.65.+a, 52.65.Kj, 91.25.Cw
Transport and amplification of a magnetic field by a flow of an electrically conducting fluid is a fundamental process in astrophysics at the planetary, stellar, and ga- lactic scales [1], as well as in laboratory plasmas, where it has been observed for a long time that a toroidal magnetic field can be sustained by applying a toroidal electric field [2]. Induced magnetic fields orthogonal to the applied ones have also been observed in flows of liquid metals:
generation of a toroidal field from an axial one by differ- ential rotation (the ‘‘!effect’’) [3–5], and generation of an electric current parallel to the applied magnetic field (the ‘‘ effect’’) [6]. These induction effects are the key mechanisms of most astrophysical and geophysical dy- namo models [7–10], as well as in the recent experimental observations of self-generation of a magnetic field by a flow of liquid sodium [11]. These experiments involve flows with geometrical constraints that are chosen in order to maximize the efficiency of the dynamo effect.
Several groups are also trying to achieve self-generation of a magnetic field in turbulent flows without, or with fewer, geometrical constraints, in order to study situ- ations that are closer to astrophysical or geophysical models [12]. It is thus of primary interest to study in- duction effects in fully developed turbulent flows. It is known that turbulent fluctuations enhance the effective resistivity [13] but other antagonistic effects such as the turbulent effect [8,9] or ‘‘ effect’’ [14] have been predicted but not experimentally observed so far.
We have measured the induced magnetic fieldBB~ gen- erated by a turbulent von Ka´rma´n swirling flow of liquid sodium submitted to a transverse external magnetic field B~
B0(see Fig. 1). The sodium flow is operated in a loop that has been described elsewhere together with the details of the experimental setup [5]. In this study, the flow is driven by rotating one of the two disks of radius R located at
position (1) or (2) in a cylindrical vessel, 410 mm in inner diameter and 400 mm in length. In most experiments presented here, we use disks of radiusR150 mm, fitted with eight straight blades of heighth10 mmdriven at a rotation frequency up to f30 Hz. Four baffles, 20 mm in height, have been mounted on the cylindrical vessel inner wall, parallel to its axis. A turbulent swirling flow with an integral Reynolds number,Re2R2f=, up to3106is driven by the rotating disk. The mean flow has the following characteristics: the fluid is ejected radially outward by the disk; this drives an axial flow toward the disk along its axis and a recirculation in the opposite direction along the cylinder lateral boundary.
The baffles inhibit the azimuthal velocity of the recircu- lating flow and thus prevent a global rotation of the fluid.
z
y x
Bo
Bz
Bx B
y
2
f1
( 1 )
f
O
S
( 2 )
FIG. 1. Geometry of the experimental setup. The flow is generated by rotating only one disk either at position (1) or (2). The magnetic field is measured at position S.
In some experiments, we have used a disk of radiusR 190 mm, fitted with 16 curved blades of height h 40 mm, with or without the lateral baffles in order to observe the effect of a stronger azimuthal flow. Without baffles, we have for the maximum velocity of the mean flowVM=2fR0:39forR150 mmand0:58forR 190 mm. Two coils generate a nearly homogeneous mag- netic field BB~0, perpendicular to the cylinder axis (see Fig. 1). The three components of the field induced by the flow are measured with a 3D Hall probe, located 200 mm away from the disk in the plane perpendicular toBB~0and containing the rotation axis. The probe distance from the rotation axis is adjustable (z42, 100, 150 mm).
The equations governing the magnetic field BB~0 B~
B~rr; t, where BB~rr; t~ is the magnetic field generated by the flow in the presence of the applied fieldBB~0, are in the MHD approximation,
r~
r BB~ 0; (1)
@ ~BB
@t
rr ~ VV~ BB~BB~0 1
0BB;~ (2) where VV~ ~rr; t is the velocity field, 0 is the magnetic permeability of vacuum, and is the fluid electric con- ductivity. The reaction of the magnetic field on the flow is characterized by the ratio of the Lorentz force to the characteristic pressure forces driving the flow. This is measured by the interaction parameter, NRB20= VM, where is the fluid density. The maximum field amplitude beingB0 12 G,Nis in the range104–102, thus the effect of the magnetic field on the flow is negli- gible. This has been checked directly by measuringBB~as a function of BB~0 at a constant driving of the flow. We calculate the mean induced fieldhBBi~ wherehistands for the average in time, as well as its rms fluctuations in time, B~
Brms. Both vary linearly with B0, thus showing that the modification of the velocity field VV~ in Eq. (2) can be neglected [5]. Thus, the only relevant dimensionless pa- rameters of our experiments are the magnetic Prandtl number, Pm 0105, where is the kinematic viscosity, and the magnetic Reynolds number, Rm 20R2f, which is proportional to the rotation fre- quencyf and has been varied up to 40 for radius of the disksR150 mm(55forR190 mm).
The three components of the mean magnetic field B~
B0 hBB~ ~rri, atz100 mm above the rotation axis, are displayed in Fig. 2 as a function of the rotation frequency.
We observe that when the rotation of the disk is reversed, f! f, we get approximately hBxi;hByi;hBzi ! hBxi;hByi;hBzi. When disk (2) is rotated instead of (1) but keepingf unchanged, we get hBxi;hByi;hBzi ! hBxi;hByi;hBzi (note that the measurements of BB~ are performed in the midplane between the two disks).
Assuming that the swirling flow has not broken the symmetries of the driving configuration, the above trans-
formations of the field components can be understood using the following symmetry transformations.
(i) The symmetry with respect to the vertical plane perpendicular toBB~0,x0z, shows that if the disk is rotated in the opposite way, f! f, we gethBxi;hByi;hBzi ! hBxi;hByi;hBzi(BB~ is a pseudovector).
(ii) The symmetry with respect to the vertical plane parallel to BB~0, y0z, followed by the transformation B~
B0! BB~0, shows that when we rotate disk (2) instead of disk (1) without changing the sign of f, we get hBxi;hByi;hBzi ! hBxi;hByi;hBzi.
The induced field component hByi at z100 mm above the rotation axis is opposed toBB~0 and increases in amplitude, thus the total field alongBB~0decreases asRmis increased. This expulsion of a transverse magnetic field from eddies is well documented, both theoretically [15]
and experimentally [16]. The expulsion is stronger close to the axis of the cylinder (z42 mm). On the contrary, closer to the cylinder lateral boundary (z150 mm), the field increases with Rm. Thus, the field is expelled from the core of the swirling flow and concentrates at its periphery. The components of the field induced perpen- dicular toBB~0 atz100 mmboth increase in amplitude from zero, reach a maximum, and then saturate whenRm is increased further. The largeRmbehavior depends onz as shown forhBxiin Fig. 3.hBxivanishes at largeRm in the core of the swirling flow (z42 mm), whereas it saturates outside (z100 and 150 mm). Figure 3 also shows that for fixedRm,hBxi increases with the distance to the cylinder axis in the range 42< z <150 mm. We can show that it should vanish for z0: indeed, the rotation of angle around the x axis followed by the transformationBB~0! BB~0, which impliesBB~ ! BB, gives~ hBxx;0; zi hBxx;0;zi.
-30 -20 -10 0 10 20 30
-0.5 0 0.5 1
f (Hz) (B o+<B >) i/B o
FIG. 2. Components of the total mean magnetic field as a function of the rotation frequency of disk (2). The disk radius is R150 mmwith straight blades. Four baffles are mounted on the inner wall of the cylindrical vessel. The magnetic field is measured atz100 mm. [()hBBxi
o ; (䊏)BohBB yi
o ; (䉱)hBBzi
o].
174501-2 174501-2
As shown in Fig. 4, hBxi and hBzi do not scale in the same way at small rotation frequency, i.e., at small Rm. The amplitude of the vertical component hBzi increases linearly, whereas the axial component hBxi increases quadratically withRm. Indeed, at the location of the measurements, near z100 mm, we observe hBxi / hBzi2 and hByi / hBzi2, roughly up to f10 Hz (see Fig. 4).
WritingBB~rr; t h~ BB~ ~rri bb~ ~rr; t, wherebb~ is the fluc- tuating part of the magnetic field, and similarly forVV, we~ get from Eq. (2) for the mean induced field
1
0hBBi ~ rr h~ VVi ~ BB~0 hVVi h~ BBi h~ vv~bbi:~ (3) When the magnetic Reynolds number is small, the first source term on the right hand side of Eq. (3) is the dominant one, and we usually get theith component of the mean induced fieldhBii /RmB0. This is the case for the vertical componenthBzi, which is mainly generated by the interaction of the toroidal velocity field withBB~0. The sign ofhBziis thus given by the one off.
However, we emphasize that bothhBxiandhByibehave quadratically at low Rm (at least at the location of our measurements). We have thus probed the nonlinear source terms of Eq. (3). As shown above, hBxx;0; zi hBxx;0;zi, andhBxiis generated by a current density parallel toBB~0. Its sign is changed by the two transforma- tions (i) and (ii) described above and is thus determined by the flow helicity,hVV~ rr ~ VV~ where the overbar stands for the spatial average. One can easily check that hBxichanges sign under any symmetry with respect to a plane containing the rotation axis, just as does the pseu- doscalarh. Although the terminology effect is usually
restricted to configurations involving scale separation, whereas our experiment does not, we do observe the generation of a current parallel toBB~0 by a cyclonic eddy as described qualitatively by Parker [7]. In addition, its magnitude increases like R2m at low Rm, and its sign is determined by the flow helicity, as in the case of the
effect.
When Rm is increased,hBxiseems to saturate forz 100 mm (see Fig. 2). Closer to the rotation axis (z 42 mm), it reaches a maximum and then decreases roughly to zero when f is increased up to 30 Hz.
This traces back to the expulsion of BB~0 from the core of the swirling flow. However, we emphasize that the effect of the expulsion on the induced fields is not straightforward. Indeed, hBzi does not vanish but satu- rates to a finite value at high frequency, even close to the rotation axis. Induction mechanisms being nonlocal, the expulsion ofBB~0 from the core of the flow does not neces- sary imply that the induced fields all decrease to zero.
It is, however, possible to enhance the effect of the expulsion by removing the baffles from the cylindrical vessel inner wall, thus allowing global rotation of the flow. Then all the components of the induced field mea- sured 100 mm away from the rotation axis decrease al- most to zero at large Rm (see Fig. 5 and compare with Fig. 2 where they stay finite). We thus observe that the axial field decreases to zero at largeRmwhen the azimu- thal flow is large enough. A similar effect has been recently computed by Ra¨dler et al. in the case of the Roberts flow [17]. These authors have shown by comput- ing terms higher than the second order inRm that the effect reaches a maximum and then tends to zero whenRm is increased further (compare their Fig. 3 with our Fig. 5).
0 50 100
0 0.2 0.4
<B x>/B o
f2 (Hz2)
0 5 10
-0.4 -0.2 0
<B z>/B o
f (Hz)
0 0.1 0.2
0 0.2 0.4
(<B z>/B
o)2
<B x>/B o
0 0.1 0.2
-0.1 0
<B y>/B o
(<B z>/B
o)2 (b) (a)
(c) (d)
FIG. 4. (a),(b) Axial and vertical mean components of the induced magnetic field as a function of the rotation frequency.
(c),(d) Axial and transverse mean components of the induced magnetic field as a function of the square of the vertical one, for f <10 Hz. Same experimental configuration as in Fig. 2 (z100 mm).
0 5 10 15 20 25 30
0 0.2 0.4 0.6 0.8 1
<B > / B0
f(Hz)
x
FIG. 3. Axial mean component of the induced magnetic field as a function of the rotation frequency of disk (2) for different depthsz: (䉱)z42 mm, (䊏)z100 mm, ()z150 mm.
Same experimental configuration as in Fig. 2.
Using dimensional analysis, we can write hBx~rri B0F~rr; Rm; Pm; N. As said above, for B0 small enough, F does not depend on the interaction parameter N.
We have provided a detailed investigation on the non- linear behavior ofF onRm even at smallRm, in a non- geometrically constrained flow. Its dependence onPm, or equivalently on the Reynolds number of the flow, is gov- erned by the relative contribution of the two non- linear source terms rr h~ VVi h~ BBi~ and rr h~ vv~bbi~ in Eq. (3). Both the velocity field, measured in water [5], and the magnetic field display large fluctuations (roughly 20%). An estimate of the effect of hvv~bbi~ can be ob- tained from the difference between the measured mag- netic field (open symbols in Fig. 5) and a numerical computation of the magnetic field induced by the mean flow alone (solid symbols) measured in a1=2-scale water experiment as explained in [18]. We observe a perfect agreement at lowRm (up to nearly f5 Hz). At larger Rm, Fig. 5 displays clear differences between the com- puted and measured fields for hBxi and hBzi. Although we cannot rule out possible inaccuracies of the nu- merical modelization (the cylindrical boundary made of copper is modeled by sodium at rest, the boundary con- ditions are periodic along thexaxis, slight nonaxisym- metry of the mean flow is not taken into account), we observe a clear difference between the magnetic induction by the turbulent flow at largeRm and the computed one resulting from the mean flow alone.
We acknowledge the assistance of J.-B. Luciani and M. Moulin. We thank J. Le´orat for letting us use his numerical code to get the curves displayed in Fig. 5.
This work is supported by the French institutions Direction des Sciences de la Matie`re and Direction de l’Energie Nucle´aire of CEA, Ministe`re de la Recherche, and Centre National de Recherche Scientifique. J.
Burguete was supported by a grant from ministerio de Educacio´n y Ciencia (Spain) while at CEA-Saclay.
*Present address: Departamento de Fı´sica y Matema´tica Aplicada, Universidad de Navarra, E-31080 Pamplona, Spain.
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‘‘Numerical Study of Homogeneous Dynamo Based on Experimental von Ka´rma´n Type Flows’’ (to be published).
0 5 10 15 20 25
0
f (Hz) (B o+<B i>)/B o
FIG. 5. Components of the total mean magnetic field as a function of the rotation frequency of disk (2). The disk radius is R190 mm with curved blades. There are no baffles on the inner wall of the cylindrical vessel. The magnetic field is measured or computed atz100 mm. The open symbols refer to the measurements and the dark ones to numerical results obtained from the computation of the induced magnetic field by the mean flow alone [(䉬)hBBxi
o ; (䊏)BohBB yi
o ; (䉱)hBBzi
o ].
174501-4 174501-4
